3.15.11 \(\int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^4} \, dx\)

Optimal. Leaf size=65 \[ \frac {44}{2401 (1-2 x)}-\frac {128}{2401 (3 x+2)}-\frac {31}{686 (3 x+2)^2}+\frac {1}{147 (3 x+2)^3}-\frac {388 \log (1-2 x)}{16807}+\frac {388 \log (3 x+2)}{16807} \]

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Rubi [A]  time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} \frac {44}{2401 (1-2 x)}-\frac {128}{2401 (3 x+2)}-\frac {31}{686 (3 x+2)^2}+\frac {1}{147 (3 x+2)^3}-\frac {388 \log (1-2 x)}{16807}+\frac {388 \log (3 x+2)}{16807} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)/((1 - 2*x)^2*(2 + 3*x)^4),x]

[Out]

44/(2401*(1 - 2*x)) + 1/(147*(2 + 3*x)^3) - 31/(686*(2 + 3*x)^2) - 128/(2401*(2 + 3*x)) - (388*Log[1 - 2*x])/1
6807 + (388*Log[2 + 3*x])/16807

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^4} \, dx &=\int \left (\frac {88}{2401 (-1+2 x)^2}-\frac {776}{16807 (-1+2 x)}-\frac {3}{49 (2+3 x)^4}+\frac {93}{343 (2+3 x)^3}+\frac {384}{2401 (2+3 x)^2}+\frac {1164}{16807 (2+3 x)}\right ) \, dx\\ &=\frac {44}{2401 (1-2 x)}+\frac {1}{147 (2+3 x)^3}-\frac {31}{686 (2+3 x)^2}-\frac {128}{2401 (2+3 x)}-\frac {388 \log (1-2 x)}{16807}+\frac {388 \log (2+3 x)}{16807}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 52, normalized size = 0.80 \begin {gather*} \frac {-\frac {7 \left (20952 x^3+29682 x^2+6887 x-2164\right )}{(2 x-1) (3 x+2)^3}-2328 \log (3-6 x)+2328 \log (3 x+2)}{100842} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)/((1 - 2*x)^2*(2 + 3*x)^4),x]

[Out]

((-7*(-2164 + 6887*x + 29682*x^2 + 20952*x^3))/((-1 + 2*x)*(2 + 3*x)^3) - 2328*Log[3 - 6*x] + 2328*Log[2 + 3*x
])/100842

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(3 + 5*x)/((1 - 2*x)^2*(2 + 3*x)^4),x]

[Out]

IntegrateAlgebraic[(3 + 5*x)/((1 - 2*x)^2*(2 + 3*x)^4), x]

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fricas [A]  time = 1.34, size = 95, normalized size = 1.46 \begin {gather*} -\frac {146664 \, x^{3} + 207774 \, x^{2} - 2328 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (3 \, x + 2\right ) + 2328 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (2 \, x - 1\right ) + 48209 \, x - 15148}{100842 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)^2/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/100842*(146664*x^3 + 207774*x^2 - 2328*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*log(3*x + 2) + 2328*(54*x^4 +
81*x^3 + 18*x^2 - 20*x - 8)*log(2*x - 1) + 48209*x - 15148)/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)

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giac [A]  time = 1.05, size = 60, normalized size = 0.92 \begin {gather*} -\frac {44}{2401 \, {\left (2 \, x - 1\right )}} + \frac {18 \, {\left (\frac {2415}{2 \, x - 1} + \frac {3038}{{\left (2 \, x - 1\right )}^{2}} + 473\right )}}{16807 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{3}} + \frac {388}{16807} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)^2/(2+3*x)^4,x, algorithm="giac")

[Out]

-44/2401/(2*x - 1) + 18/16807*(2415/(2*x - 1) + 3038/(2*x - 1)^2 + 473)/(7/(2*x - 1) + 3)^3 + 388/16807*log(ab
s(-7/(2*x - 1) - 3))

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maple [A]  time = 0.01, size = 54, normalized size = 0.83 \begin {gather*} -\frac {388 \ln \left (2 x -1\right )}{16807}+\frac {388 \ln \left (3 x +2\right )}{16807}+\frac {1}{147 \left (3 x +2\right )^{3}}-\frac {31}{686 \left (3 x +2\right )^{2}}-\frac {128}{2401 \left (3 x +2\right )}-\frac {44}{2401 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)/(1-2*x)^2/(3*x+2)^4,x)

[Out]

1/147/(3*x+2)^3-31/686/(3*x+2)^2-128/2401/(3*x+2)+388/16807*ln(3*x+2)-44/2401/(2*x-1)-388/16807*ln(2*x-1)

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maxima [A]  time = 0.63, size = 56, normalized size = 0.86 \begin {gather*} -\frac {20952 \, x^{3} + 29682 \, x^{2} + 6887 \, x - 2164}{14406 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} + \frac {388}{16807} \, \log \left (3 \, x + 2\right ) - \frac {388}{16807} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)^2/(2+3*x)^4,x, algorithm="maxima")

[Out]

-1/14406*(20952*x^3 + 29682*x^2 + 6887*x - 2164)/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8) + 388/16807*log(3*x + 2
) - 388/16807*log(2*x - 1)

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mupad [B]  time = 1.08, size = 46, normalized size = 0.71 \begin {gather*} \frac {776\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{16807}-\frac {\frac {194\,x^3}{7203}+\frac {1649\,x^2}{43218}+\frac {6887\,x}{777924}-\frac {541}{194481}}{x^4+\frac {3\,x^3}{2}+\frac {x^2}{3}-\frac {10\,x}{27}-\frac {4}{27}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)/((2*x - 1)^2*(3*x + 2)^4),x)

[Out]

(776*atanh((12*x)/7 + 1/7))/16807 - ((6887*x)/777924 + (1649*x^2)/43218 + (194*x^3)/7203 - 541/194481)/(x^2/3
- (10*x)/27 + (3*x^3)/2 + x^4 - 4/27)

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sympy [A]  time = 0.16, size = 54, normalized size = 0.83 \begin {gather*} \frac {- 20952 x^{3} - 29682 x^{2} - 6887 x + 2164}{777924 x^{4} + 1166886 x^{3} + 259308 x^{2} - 288120 x - 115248} - \frac {388 \log {\left (x - \frac {1}{2} \right )}}{16807} + \frac {388 \log {\left (x + \frac {2}{3} \right )}}{16807} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)**2/(2+3*x)**4,x)

[Out]

(-20952*x**3 - 29682*x**2 - 6887*x + 2164)/(777924*x**4 + 1166886*x**3 + 259308*x**2 - 288120*x - 115248) - 38
8*log(x - 1/2)/16807 + 388*log(x + 2/3)/16807

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